The RPA has been widely used in the existing literature to optimize the passive reflection coefficients of IRS (see, e.g., [1]}, [2]}, [3]}, [4]}) without assuming any knowledge of the CSI. This algorithm is also applicable to our considered integrated IRS-BS system for solving (P1). Specifically, a given number of feasible solutions of \(\mbox{{$ \Theta  $}}\)  are randomly generated at the BS, where the phase shift of each reflecting element in \(\mbox{{$ \Theta  $}}\)  is chosen from interval \([0,2\pi )\)  uniformly and independently, and then the solution with the maximum objective value of (P1) is selected as the final solutionIn practice, with each realization of \(\mbox{{$ \Theta  $}}\) , the BS needs to estimate the effective channels from the users to the antenna array of the BS, i.e., \(\lbrace \mbox{{$ h $}}_{k}(\mbox{{$ \Theta  $}}),~k\in \mathcal {K}\rbrace \) , by applying the channel estimation schemes for the conventional multi-antenna BS without integrated IRS.. Note that this algorithm is in fact used for the initialization of Algorithm REF  assuming perfect CSI at the BS. The performance of RPA is highly dependent on the phase control of the single-reflection and double-reflection channel components with independent reflection coefficient realizations. However, for the double-reflection channel components, due to the coupled terms, i.e., \(\mbox{{$ g $}}_{k,j,q}^{n_{j},n_{q}}\vartheta _{j,n_{j}}\vartheta _{q,n_{q}},~j\ne q\in \mathcal {J}, n_{j}\in \mathcal {N}_{j}, n_{q}\in \mathcal {N}_{q}\)  in (REF ), independent phase control with randomly generated reflection coefficients for all IRS simultaneously becomes impossible, which may result in the performance loss of RPA. Thus, we use RPA as a benchmark algorithm and propose an improved solution in the following.
